论文标题
抛物线最佳控制问题与组合开关约束 - 第二部分：外近似算法
Parabolic optimal control problems with combinatorial switching constraints -- Part II: Outer approximation algorithm
论文作者
论文摘要
我们考虑了部分微分方程的最佳控制问题，其中控件采用二进制值但在时间范围内有所不同，因此可以将它们视为动态开关。开关模式可能会受到组合约束，例如，开关总数上的上限或两个开关之间的时间下限。在同伴论文[arxiv：2203.07121]中，我们描述了可行开关模式的凸壳的$ l^p $ clublosure the coblesible toptions hull hull hull hull hull hull hull hull的壳体壳体的相交是从有限维透射得出的凸面集的相交。在本文中，所得的外部描述用于构建功能空间中的外近似算法，该迭代被证明可以在$ l^2 $中以$ l^2 $的强烈收敛，以汇总到凸的最佳控制问题的全局最小化器。在外部近似算法的每种迭代中产生的线性季度子问题都是通过半平滑牛顿方法求解的。两个空间维度中的数值示例说明了总体算法的效率。
We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [arXiv:2203.07121], we describe the $L^p$-closure of the convex hull of feasible switching patterns as intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in $L^2$ to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semi-smooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.
